Properties

Label 139.1.b.a
Level 139
Weight 1
Character orbit 139.b
Self dual Yes
Analytic conductor 0.069
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM disc. -139
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 139 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 139.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.069370036756\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.139.1
Artin image size \(6\)
Artin image $S_3$
Artin field Galois closure of 3.1.139.1

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut +\mathstrut q^{16} \) \(\mathstrut -\mathstrut q^{20} \) \(\mathstrut -\mathstrut q^{28} \) \(\mathstrut -\mathstrut q^{29} \) \(\mathstrut -\mathstrut q^{31} \) \(\mathstrut +\mathstrut q^{35} \) \(\mathstrut +\mathstrut q^{36} \) \(\mathstrut +\mathstrut 2q^{37} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut q^{44} \) \(\mathstrut -\mathstrut q^{45} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut -\mathstrut q^{52} \) \(\mathstrut +\mathstrut q^{55} \) \(\mathstrut -\mathstrut q^{63} \) \(\mathstrut +\mathstrut q^{64} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut -\mathstrut q^{71} \) \(\mathstrut +\mathstrut q^{77} \) \(\mathstrut -\mathstrut q^{79} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut +\mathstrut q^{81} \) \(\mathstrut -\mathstrut q^{83} \) \(\mathstrut -\mathstrut q^{89} \) \(\mathstrut +\mathstrut q^{91} \) \(\mathstrut -\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/139\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
138.1
0
0 0 1.00000 −1.00000 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
139.b Odd 1 CM by \(\Q(\sqrt{-139}) \) yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(139, [\chi])\).